CLEVELAND -- When Jason Giambi penned his name on a new contract with the Indians this week, the veteran put himself in a position to have two decades of big league baseball on his resume. Even after all he has accomplished, Giambi finds that hard to fathom.

Twenty years.

"I think you only dream about it," Giambi said on Friday. "You go through your certain stepping stones of your career, where it's, 'I need to get to the big leagues,' and then, 'I need to stay in the big leagues,' and then, 'I need to try to last in the big leagues and get my 10 years in.'

"Then, everything is kind of gravy after that. So I'm definitely blessed and I'm truly grateful for this opportunity. Not in my wildest dreams did I think I'd still be here."

Here is Cleveland, and the 42-year-old Giambi insists there is no place he would rather be in 2014. That is why on Thursday -- the first day eligible players could file for free agency -- he took himself off the open market by signing a Minor League contract to return to the Indians. Giambi will head into Spring Training as a non-roster invitee, but the Opening Day roster will surely have a spot tentatively reserved.

During this past season, the Tribe brought Giambi into the fold as a part-time designated hitter, pinch-hitter and clubhouse leader. He went from a managerial candidate with the Rockies to a leader for the Indians, delivering in critical situations off the bench with his bat and keeping the team focused within the clubhouse with his words.

Whether Giambi wanted to move forward as a player or coach, Cleveland made it crystal clear that it wanted him back for at least one more year.

"We would like to continue our relationship with G," manager Terry Francona said at the end of the season, "probably as long as he would like to."

Giambi is not ready to close his bat bag for good just yet.

"I told them I'd really love to play," Giambi said.

And Giambi is not sure he will ever definitively declare any season his last.

"I don't know if there will ever come a time that I'll say that," Giambi said. "But I think your body will just give you a point and say, 'I just can't do it anymore.' There is going to come a point."

Part of Giambi's drive to keep playing surely stems from the fact that a World Series ring has eluded him in each of his past 19 seasons in the big leagues. He reached the Fall Classic with the Yankees in 2003, but New York lost in six games to the Marlins. This past season, the Indians won 92 games and clinched the top American League Wild Card spot, only to lose to the Rays in the AL Wild Card Game.

Giambi believes Cleveland is on the cusp of contending for multiple years.

"I think what we have here in Cleveland is something special," Giambi said. "I think this is a team that can have an opportunity to win a World Series. We definitely have the pitching to do it. We have the offense to make those strides. We're growing as a team, and I think we're going to keep getting better and better."

Giambi is not the offensive force he was in his prime with the A's and Yankees, but the aging slugger -- with 438 career home runs under his belt -- was still a presence at the plate for the Tribe in 2013. He hit only .183 in 71 games, but Giambi posted a 1.181 OPS in the ninth inning, averaged one RBI per six at-bats and hit at a .271 clip (.960 OPS) with runners in scoring position.

Giambi launched three pinch-hit home runs and twice established a Major League record as the oldest player to hit a walk-off home run. Hank Aaron held that mark since 1976 until Giambi's walk-off shot against the White Sox on July 29, and then the DH broke his own record with a pinch-hit, two-run walk-off blast against White Sox closer Addison Reed on Sept. 24.

That was the kind of blast Indians fans had dreamt about.

As for Giambi's 20th season in the Majors, that dream is becoming very real.

"It's kind of funny," Giambi said. "They used to call me 'kid' all the time when I was playing [in Oakland]. Now, I'm calling [Jason] Kipnis that. It gets passed on. I really think that's the gift of the game. You pass it back down, and even when you're out of this game, that's how you still live in the game."